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In contrast with rich investigations about the translation of an antiferromagnetic (AFM) texture, spin precession in an AFM texture is seldom concerned for lacking an effective driving method. In this work, however, we show that under an alternating spin-polarized current with spin along the AFM anisotropy axis, spin precession can be excited in an AFM DW. Especially, chaotic spin precession occurs at moderate interfacial Dzyaloshinskii-Moriya interaction (DMI), which contributes to a nonlinear term in the dynamic equation of DW precession. Also, crisis-induced intermittent chaos appears when the current density is higher than a critical value. This work not only paves a way to unravel rich spin precession behaviors in an AFM texture but also provides guidelines for developing ultrafast spintronic devices based on new physical principles.